Tools for the Analysis and Design of Complex Multi-Scale Networks: Dynamics; Security; Uncertainty MURI Annual Review Columbus OH, October 14, 2010 J. Walrand, PI

Contents: Dynamics of Local Protocols Security of Graphs Games with Uncertainty

Dynamics of Local Protocols: Scientific Objectives Approach: An Example General Approach Example 1 Example 2

Scientific Objectives Researchers propose new protocols for networks The protocols are based on local interactions These protocols are difficult to evaluate New methodology for Delays Transients Impact of mobility

Approach: An Example Link 1 Link 2 Link 3

Approach: An Example (continued)

Approach: An Example (continued)

Approach: An Example (continued) How to evaluate delays, mobility, transients? Service rate of queue 1 = P({1}) + P({1, 3}) => Service rate of queues = f(R) = f(q)

Approach: An Example (continued) How to evaluate delays, mobility, transients?

General Approach: Local interactions, randomized protocols Use fluid limit approximation to get ODEs Evaluate ODEs for transient analysis, delays, mobility

Example 1: ODEs vs. Simulations

Example 2: ODEs vs. Simulation

Example 2: ODEs + Simulations with Virtual Arrivals

Security of Graphs: Model Theorem Examples Application Algorithm

Model: Graph Goal of Network Manager: Choose a Spanning Tree T to minimize Probability of Attack Goal of Attacker: Attack a Link L to maximize Probability that L is in T Attacker: + 1 Manager: - 1 One-Shot Zero-Sum Game (Simultaneous Actions)

Theorem: There is a Nash Equilibrium where Attacker attacks only the links of a critical set C, with equal probabilities Manager chooses only trees that have a minimal intersection with C, and have equal likelihood of using each link of C, not larger than that of using any link not in C. [Such a choice is possible.] There is a polynomial algorithm to find C [Cunningham] Critical Set: C if it has a maximal vulnerability ν(C) where

Example 1: Critical Set: C if it has a maximal vulnerability ν(C) where ν(C) = 4/7 Every spanning tree must use at least 4 links of C, i.e., 4 out of 7 links of C. This fraction is maximal for C. Note that the fraction is only ½ for the minimal cut set.

Example 1: ν(C) = 4/7 Nash equilibrium: Attacker attacks each link of C with probability 1/7; Manager chooses spanning trees that use only 4 links of C, in such a way that the probability that a chosen tree uses a particular link of C is 4/7, and this probability is not larger than for any link not in C.]

Example 2: Critical Set: C if it has a maximal vulnerability ν(C) where ν(C) = 1/2 Every spanning tree must use at least 1 link of C, i.e., 1 out of 2 links of C. This fraction is maximal for C. In this case, C is the minimal cut set.

Example 2: ν(C) = 1/2 Nash equilibrium: Attacker attacks each link of C with probability 1/2; Manager chooses spanning trees that use only 1 link of C, in such a way that the probability that a chosen tree uses a particular link of C is ½, which is not larger than the probability of using any link not in C.

Application: Network Design: How to choose a topology with minimal vulnerability? V(G) = 3/4 V(G) = 3/5 V(G) = 2/3 2/3 > 3/5

Games with Uncertainty: Excess of Caution can Hurt Should one be Optimistic?

Excess of Caution can Hurt Relay Network:

Excess of Caution can Hurt Relay Network:

Optimism Best Response: Consistency: Attitude:

Optimism Attitude Game: Example: Consistent Sets: Attitude Game: